// bayesian latent trait model 4 for "Estimating Smooth Panels"
// item intercepts / biases, no slopes / factor loadings 
// single autoregressive error
// responses are modelled as beta-binomial count
// validation test

data {
  int<lower=1> N;		// number of national survey opinions
  int<lower=1> J;		// number of countries
  int<lower=1> K;		// number of items 
  int<lower=1> T;		// number of years
  int<lower=1,upper=J> jj[N];		// country j for opinion n
  int<lower=1,upper=K> kk[N];		// item k for opinion n
  int<lower=1,upper=T> tt[N];		// year t for opinion n
  int<lower=1> x[N];		// vector of survey responses, count
  int<lower=1> samp[N];		// vector of sample sizes
}

parameters {
  real<lower=0> sigma_theta;		// opinion evolution error variance  
  real<lower=0> sigma_lambda;		// item intercepts error variance
  vector[K-1] lambda_raw;		// raw item intercepts
  matrix[T,J] theta_raw;		// raw matrix of T by J latent traits
  row_vector[J] theta_init;		// initial latent traits for first year
  real mu_lambda;		// mean of item intercepts
  real<lower=0> phi;		// dispersion parameter
}

transformed parameters {
  matrix[T,J] theta;		// matrix of T by J latent traits
  vector[N] theta_tt_jj;		// N-vector for expanded theta vales
  vector<lower=0,upper=1>[N] eta;		// fitted values, on logit scale
  vector<lower=0>[N] alpha;		// beta lambda parameter
  vector<lower=0>[N] beta;		// beta beta parameter  
  vector[K] lambda;		// item intercepts
  theta[1] = theta_init; 
  lambda[1] = 1;		// constrain initial intercept
  lambda[2:K] = mu_lambda + sigma_lambda * lambda_raw[1:K-1];	
  for (t in 2:T)		// parameter expansion for theta
	theta[t] = theta[t-1] + sigma_theta * theta_raw[t-1];
  for (i in 1:N) 	
  	theta_tt_jj[i] = theta[tt[i], jj[i]];	// expand theta to N-vector
  eta = inv_logit(lambda[kk] + theta_tt_jj);	// fitted values model
  alpha = phi * eta;		// reparamaterise beta-binom lambda par
  beta = phi * (1 - eta);		// reparamaterise beta-binom beta par
}

model {
  x ~ beta_binomial(samp, alpha, beta); 		// response model
  phi ~ gamma(4, 0.1);
  sigma_lambda ~ cauchy(0, 2); 
  sigma_theta ~ cauchy(0, 2); 
  mu_lambda ~ normal(1, 2);
  theta_init ~ normal(0, 1);
  lambda_raw ~ normal(0, 1);
  for(t in 1:T-1) 
	theta_raw[t] ~ normal(0, 1);
}
